| L(s) = 1 | + 2-s + 2.65·3-s + 4-s + 3.01·5-s + 2.65·6-s + 4.30·7-s + 8-s + 4.06·9-s + 3.01·10-s − 5.94·11-s + 2.65·12-s − 1.92·13-s + 4.30·14-s + 8.02·15-s + 16-s + 1.43·17-s + 4.06·18-s + 7.00·19-s + 3.01·20-s + 11.4·21-s − 5.94·22-s − 7.57·23-s + 2.65·24-s + 4.10·25-s − 1.92·26-s + 2.84·27-s + 4.30·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 0.5·4-s + 1.34·5-s + 1.08·6-s + 1.62·7-s + 0.353·8-s + 1.35·9-s + 0.954·10-s − 1.79·11-s + 0.767·12-s − 0.534·13-s + 1.14·14-s + 2.07·15-s + 0.250·16-s + 0.347·17-s + 0.958·18-s + 1.60·19-s + 0.674·20-s + 2.49·21-s − 1.26·22-s − 1.58·23-s + 0.542·24-s + 0.821·25-s − 0.378·26-s + 0.546·27-s + 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.287046939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.287046939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 + 5.94T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 7.00T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 + 3.05T + 37T^{2} \) |
| 41 | \( 1 - 6.67T + 41T^{2} \) |
| 43 | \( 1 - 5.60T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 + 3.99T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.159093203500979633110660420664, −7.70155145220020675179428003706, −7.43589096937475260141449872303, −5.87329166498938326177513313029, −5.37620283602010963839597522572, −4.80345808049549854904438372451, −3.71542142805090533500847396454, −2.76062809601740873062364481424, −2.13363638399441961011163115706, −1.64414842772641822120046036837,
1.64414842772641822120046036837, 2.13363638399441961011163115706, 2.76062809601740873062364481424, 3.71542142805090533500847396454, 4.80345808049549854904438372451, 5.37620283602010963839597522572, 5.87329166498938326177513313029, 7.43589096937475260141449872303, 7.70155145220020675179428003706, 8.159093203500979633110660420664
